TS EAMCET · Maths · Definite Integration
\(\int_0^{\frac{\pi}{2}} \frac{x \tan x \sec ^2}{\tan ^4 x+1} d x=\)
- A \(\frac{\pi^2}{16}\)
- B \(\frac{\pi^2}{4}\)
- C \(\frac{\pi^2}{8}\)
- D \(\frac{\pi^2}{32}\)
Answer & Solution
Correct Answer
(D) \(\frac{\pi^2}{32}\)
Step-by-step Solution
Detailed explanation
Let \(I=\int_0^{\frac{\pi}{2}} \frac{x \tan x \sec ^2 x}{\tan ^4 x+1} d x\) ...(i) \(=\int_0^{\frac{\pi}{2}}\left(\frac{\pi}{2}-x\right) \frac{\cot x \cdot \operatorname{cosec}^2 x}{\cot ^4 x+1} d x\)…
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